Find the parametric equations of the line tangent to the curve of intersection of the paraboloid
z = x² + y² and the ellipsoid 4x² + y² + z² = 9 at the point ( -1, 1, 2 ).
Probable use of the gradient vector (as this is the chapter we are in)
The Attempt at a Solution
I have found what I believe to be the curve of intersection by setting the equations of the surfaces equal to each other:
x² + y² - z = 4x² + y² + z² - 9
9 = 3x² + z² + z
F(x,y,z) = 3x² + z² + z - 9 (as a function)
And the corresponding gradient vector:
< 6x, 0, 2z + 1 >
But I cannot figure out how to convert this from cartesian to polar (or if I even need to) and get the correct answer. I've tried several paths (i.e. substituting x=rcosσ) but my answer never matches the expected answer, which contains no trigonometry. Does the correct approach involve substituting x² + y² for z? This seems to be the most logical approach but I haven't been able to apply it correctly.